ARITHMETICS OF JORDAN ALGEBRAS

27

6 -p

•7

a)

(the split quaternion case) .

In both c a s e s M(C . J ) = U(C ) = W(C , J, ). All symmetric elements of

n* 7 n n 1

(C

3

J ) have diagonal entries in K except when C is a split quaternion algebra

n 7

and K has characteristic 2. In that cas e Jf(C, -) has dimension 3. The

algebra obtained by considering all symmetric elements of (C , J ) is an

isotope of M($ , T ), T the transpos e involution. In that sens e it is not

really a split algebra. Moreover it does not seem to yield to our methods

unles s the bas e field is assumed to be complete discrete . That cas e will be

treated in §5.

If C =K © K, $ =M(C ) s K . By Proposition 1, maximal orders of

n n

K coincide with maximal orders of K . But thes e are of the form E(L), L

n n

an o-lattice of an n-dimensional K-vector spac e If. An isomorphism of

maximal orders induces either an automorphism K or an antiautomorphism

of K . Any antiautomorphism can be obtained by composing the transpos e

antiautomorphism with an automorphism. By Lemma 3

L = ox © . . . 0 ox 0 ax , E(L) is isomorphic to

a \

and the transpos e sends thi s order

-1

Q 0

onto an order isomorphic to E(L') where L1 = ox ox . 0 0 x .

n- 1 n